When ready to bake, preheat oven to 375 degrees and position racks to upper and lower thirds of the oven. Brush the pastries all over with remaining beaten egg white. Bake until golden brown, about 25 minutes, rotating the baking sheets from front to back and top to bottom halfway through the baking time.
As soon as the baking sheets come out of the oven and the pastries are piping hot, do something that seems a little crazy. Compress each pastry slightly by using the baking sheet with the other pastry on it — just set the pan on top of the pastry, and press gently to eliminate the air pocket between the pastry and filling. Allow the pastries to cool completely on the pans set over wire racks.
To prepare icing: In a small bowl, combine 1 cup powdered sugar and pinch of salt. Whisk in the vanilla extract and just enough water, 1 teaspoon at a time (about 4), to form a smooth, opaque glaze — when you lift the whisk, it should easily form a fluid ribbon without breaking, but doesn't drip in little droplets in a runny fashion (it's often just a matter of a few extra drops of water from your fingertips to get the consistency just right, so don't add too much water in the beginning). Using a small, flexible spatula, ice the kringles with icing. Let the icing set before slicing and serving.
Nutrition information per each of 24 servings (about 2 inches each):
Calories196Fat12 gSodium100 mg
Carbohydrates20 gSaturated fat6 gAdded sugars11 g